7 papers · 1 filter
Counterexamples to the Landis conjecture in dimensions three and higher
Rupert L. Frank, Paata Ivanisvili
In every dimension three and higher we construct a nonzero function that satisfies the stationary Schrödinger equation with bounded, real-valued potential and decays like $\exp(-c…
Pleijel's theorem for a class of degenerate elliptic operators
Rupert L. Frank, Bernard Helffer
We prove an asymptotic upper bound on the number of nodal domains of eigenfunctions of a class of degenerate elliptic operators. Our proof yields the same constant as in Pleijel's…
The distance between homotopy classes of Sobolev maps on spheres
Rupert L. Frank, Paata Ivanisvili
We consider self-maps of a sphere in the critical Sobolev space with a given Brouwer degree. Our main result is that the (directed) distance between maps of different degrees is eq…
Minimizers for Coulomb gases constrained to a halfspace
Rupert L. Frank, Paata Ivanisvili, Clara Torres-Latorre
We consider a family of optimization problems, based on a mean-field description of particles interacting through Coulomb forces in a quadratic trap. In addition, the particles are…
The sharp log Sobolev inequality on finite cycles
Rupert L. Frank, Paata Ivanisvili
We settle the problem of finding the sharp constant in the log Sobolev inequality on the -cycle for all , by showing that it is equal to half of the spectral gap. We ded…
Uniform bounds for Neumann heat kernels and their traces in convex sets
Rupert L. Frank, Simon Larson
We prove a bound on the heat trace of the Neumann Laplacian on a convex domain that captures the first two terms in its small-time expansion, but is valid for all times and depends…